We characterize a near-infrared C-RED ONE camera from First Light Imaging (FLI). This camera uses a SAPHIRA electron avalanche photo-diode array (e-APD) from Leonardo (previously Selex). To do so, we developed a model of the signal distribution. This model allows a measurement of the gain and the Excess Noise Factor (ENF) independently of preexisting calibration such as the system gain. The results of this study show a gain which is 0.53 ± 0.04 times the gain reported by the manufacturer. The measured ENF is 1.47 ± 0.03 when we expected 1.25. For an avalanche gain of ' 60 and a frame rate larger than 100 Hz, the total noise can be lower than 1 e-/frame/pixel. The lowest dark current level is 90e-/s/pixel, in agreement with the expected H-band background passing through the camera window. These performance values provide a significant improvement compared to earlier-generation PICNIC camera and allowed us to improve the performance of the Michigan infrared combiner (MIRC) instrument at the Center for High Angular Resolution Astronomy (CHARA), as part of our MIRC-X instrumentation project.